radiance equation

7/29/2019 Radiance Equation

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The Radiance Equation

Mel Slater


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Outline

Introduction

Light

Simplifying Assumptions Radiance

Reflectance


Traditional Rendering Solutions

Visibility

Conclusions


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Introduction

Lighting is the central problem of real-time

graphics rendering

Arbitrary shaped lights

Changes in lighting conditions

Real-time shadows

Real-time reflections

Mixtures of many different types of surface


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Introduction

Real-time walkthrough with global illumination

Possible under limited conditions

Radiosity (diffuse surfaces only)

Real-time interaction

Not possible except for special case local

illumination

Why is theproblem so hard?


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Light

Visible light is electromagnetic radiation

with wavelengths approximately in the

range from 400nm to 700nm

400nm 700nm


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Light: Photons

Light can be viewed as wave orparticlephenomenon

Particles arephotonspackets of energy which travel in a straight line

in vaccuum with velocity c (300,000m.p.s.)

The problem of how light interacts withsurfaces in a volume of space is an exampleof a transportproblem.


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Light: Radiant Power

denotes the radiant energy orflux in a volumeV.

The flux is the rate of energy flowing through asurface per unit time (watts).

The energy is proportional to the particle flow,since each photon carries energy.

The flux may be thought of as the flow of photonsper unit time.


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Light: Flux Equilibrium

Total flux in a volume in dynamic

equilibrium

Particles are flowing

Distribution is constant

Conservation of energy

Total energy input into the volume = totalenergy that is output by or absorbed by matter

within the volume.


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Light: Fundamental Equation

Input

Emissionemitted from within volume

Inscatteringflows from outside Output

Streamingwithout interaction with matter in the

volume

Outscatteringreflected out from matter

Absorptionby matter within the volume

Input = Output


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Light: Equation

(p,) denotes flux atpV, in direction

It is possible to write down an integral equation

for

(p,

) based on: Emission+Inscattering = Streaming+Outscattering +Absorption

Complete knowledge of(p,) provides a

complete solution to the graphics renderingproblem.

Rendering is about solving for(p,).


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Simplifying Assumptions

Wavelength independence

No interaction between wavelengths (nofluorescence)

Time invariance Solution remains valid over time unless scene changes

(nophosphorescence)

Light transports in a vacuum (non-participating

medium) free space interaction only occurs at the surfaces of

objects


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Radiance

Radiance (L) is the flux that leaves a

surface, per unit projected area of the

surface, perunit solid angle of direction.

n

dA

L

d = L dA cos d


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Radiance

For computer graphics the basic particle is not

the photon and the energy it carries but the ray

and its associated radiance.

n

dA

L d

Radiance is constant along a ray.


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Radiance: Radiosity,

Irradiance Radiosity - is the flux per unit area that

radiates from a surface, denoted byB.

d = B dA

Irradiance is the flux per unit area that

arrives at a surface, denoted byE.

d = E dA


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Radiosity and Irradiance

L(p,) is radiance at p in direction

E(p,) is irradiance at p in direction

E(p,) = (d/dA) = L(p,) cos d


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Reflectance BRDF

Bi-directional

Reflectance

Distribution Function

Relates

Reflectedradiance toincomingirradiance

ir

Incident ray

Reflected ray

Illumination hemisphere

f(p, i ,r)


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Reflectance: BRDF

Reflected Radiance = BRDFIrradiance

L(p, r) = f(p,i ,r) E(p, i )

= f(p,i ,r) L(p, i ) cosi di

In practice BRDFs hard to specify

Rely on ideal types

Perfectly diffuse reflection

Perfectly specularreflection

Glossy reflection

BRDFs taken as additive mixture of these


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Radiance L(p, ) at a point p in direction is the sum of

Emitted radiance Le(p, )

Total reflected radiance

Radiance = Emitted Radiance + Total Reflected Radiance


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The Radiance Equation:

Reflection Total reflected radiance in direction :

f(p,

i,

) L(p,

i) cos

id

i

Radiance Equation:

L(p, ) = Le(p, ) +

f(p,

i,

) L(p,

i) cos

id

i

(Integration over the illumination hemisphere)


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The Radiance Equation p is considered to be on a surface, but can be

anywhere, since radiance is constant along a ray,

trace back until surface is reachedat p, then

L(p, i) = L(p, i )

p*

i

pL(p, )

L(p, ) depends on allL(p*, i) which in turnare recursively defined.

The radiance equation models global illumination.


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Traditional Solutions to the

Radiance Equation The radiance equation embodies totality of

all 2D projections (view).

Extraction of a 2D projection to form animage is called rendering.


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Traditional Solutions

Local

Illumination

Global

Illumination

View

Dependent

Real time

graphics:

OpenGL

Ray tracing

Path tracing

View

Independent

Flat shaded

graphics

(IBR)

Radiosity

(Photon

Tracing)


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(Image Based Rendering)

IBR not a traditional solution

Images for a new view constructed from a

large collection of existing images

No lighting computations at all.

Light Field Rendering specific instance to

be discussed later.


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Visibililty Where does an incident ray through the image

plane come from?

Which surface?

Ray tracing in principle has to search all surfacesfor possible intersections

Radiosity has to include visibility in form-factorcalculations between surfaces

Real-time rendering solves visibility problem on apixel by pixel basis (z-buffer).

Major complication for large scenes

We will see later that LFR does not have this

visibility problem.


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Conclusions

Graphics rendering is concerned with solution ofintegral radiance equation

Traditional solutions are various kinds of

approximations to this equation. Rendering is the process ofextracting images

from the equation.

Rendering may be view dependent orindependent,

together with a global orlocal illuminationsolution.

radiance equation

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